As the flexible engagement gear device, such types are known whose component part, flexible external gear is of a circular shape, a cup shape, and a silk hat shape. Among these types of flexible engagement gear devices, the applicant of this application has proposed a flexible engagement gear device having a flat silk-hat flexible external gear in JP-A 8-166051 and JP-A 8-166052. Using a flat silk-hat flexible external gear makes it possible to constitute a flexible engagement gear device flat.